Upcoming Talks

May 22, 2019

Moments [with] L-functions and Class Numbers

In this talk we investigate the moments and the distribution of $L(1,\chi_D)$, where $\chi_D$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_q$, as $n\to\infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\chi_D)$ in a large uniform range. Previously, only the first moment had been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\chi_D)$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1, \chi_D)$, that is not present in the number field setting. We also obtain $\Omega$-results for the extreme values of $L(1,\chi_D)$, which we conjecture to be best possible.

Specializing $n=2g+1$ and making use of one case of Artin's class number formula, we obtain similar results for the class number $h_D$ associated to $\mathbb{F}_q(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin's class number formula and deduce analogous results for $h_DR_D$ where $R_D$ is the regulator of $\mathbb{F}_q(T)[\sqrt{D}]$.

Past Talks

May 8, 2019

How to make a really good smoothie

Blenders have become ubiquitous household kitchen items. Despite their prevalence and R&D money invested into their design, relatively little is known of their actual workings and how smoothies are made. In this talk, I'll present a bottom-up tutorial style introduction to the fluid mechanics of what happens inside a blender.

In the spirit of classical applied mathematics, I will begin with an overview of the Navier-Stokes equations of fluid mechanics, demonstrate a non-dimensionalisation procedure, and show how geometric and physical arguments can help to find an analytic solution to a highly idealised flow in the blender. I'll also talk about some strategies that can be used to optimise your smoothie making at home.

This will be a tutorial style talk and will touch on several aspects of classical applied mathematics. This work is the result of a three day long mathematical modelling camp at the University of Oxford, which took place in April, 2019.

May 15, 2019

Modelling the risk of cardiovascular events due to influenza infection

We develop a dynamic model which incorporates the immune response, inflammatory system, and blood coagulation to understand the assocation of cardiac events with multiple biological pathways a human host. We synthesize these biological systems and integrate them into a cohesive modelling framework to study their connections to blood clotting. This is based on an on-going project of the NSERC/Sanofi Industrial Research Chair Program "Vaccine Mathematics, Modelling and Manufacturing."

May 1, 2019

Pure States and Where to Find Them

In this talk, we will introduce the notion of pure states from both the representation-theoretic point of view and as a generalization of the evaluation functionals. Alos, we will discuss a long-standing open problem related to pure states of C*-algebras, often known as Naimark's Problem. If time allows, we will also give a taste on the role that some set-theoretic machinery play; from some classical infinitary combinatorics to the application of the (famous for being infamous) forcing technique.

April 24, 2019

The Combinatorics of Free Probability

Free Probability involves the study of random variables which do not necessarily commute. In this talk, we will focus on the key notion of free independence and its combinatorial interpretation which implements lattices of partitions on finite sets. We will explore how the combinatorics of free probability yield instances of objects that are of interest within the framework of Operator Algebras and discuss some of the current lines of research.

April 17, 2019

Semidefinite Programming Approach for Modelling Hard Discrete Optimization Problems

In this talk, I will review teh basic concepts of Semidefinite Programming (SDP) and demonstrate some of its applications to the areas of mathematical programming and combinatorial optimization. In particular, a novel SDP-based formulation for the subset-sum knapsack problem as well as the cutting stock problem will be proposed, and it will be shown that the bounds obtained by relaxing the constraints of the proposed mathematical models could effectively dominate the existing linear relaxation bounds.

April 10, 2019

Reducing NOx levels: Effectiveness of Titanium Oxide coatings on cement for photo-catalytic oxidation of traffic-related air pollution

Worldwide, countries find traffic-related air pollution (TRAP) extremely difficult to control. In Canada, approximately one third of the population lives within 500 metres of highways and within 100 metres of busy roads. As a result, 10 million Canadians are at risk of major health issues due to TRAP. Each year in Canada, approximately 21,000 premature deaths are attributed to air pollution. In this talk, we will look at the effectiveness of using titanium oxide coatings on raods to reduce TRAP.

This project was undertaken at the Industrial Problem Solving Workshop in May 2018 in Ottawa jointly hosted by the National Research Council (NRC), so I will also talk about my experience there.

April 3, 2019

An application of computational topology to ECG analysis

In this talk we will give a brief introduction to simplicial homology and some methods used in computational topology. We will then consider an application to time-delay lifts of periodic functions and ECG signals.

March 27, 2019

A Mathematicians Guide to Finance - How I stopped worrying and learned to love financial economics

Mathematical Finance typically is not overly concerned with standard finance theory or economics generally. Rather, practitioners and academics usually focus on modelling an endogenous process (e.g. a stock price) apply a simple modelling assumption (e.g. no arbitrage) and derive formulae useful for valuation (e.g. the Black-Scholes-Merton equation). Along with studies in the actuarial sciences this is often the extent of mathematicians interaction with finance. In this talk we will explore a stylized history of modern financial economics and the relevant mathematical problems that arise in fiance and insurance with some discussion of open problems of interest.

March 20, 2019

Almost Disjoint Families and Topology

We will discuss some work of Michael Hrusak on almost disjoint families. We will prove a frew of the results in this paper with a small introduction to Luzin families. The talk will end with starting some interesting open problems which remain in this area.

November 26, 2018

On Efimov's Problem

Every topological space is infinite and Hausdorff (just while you read this, of course)

Question 1: Does Every compact space contain an infinite convergent sequence?

To see why the answer to Question 1 is surprisingly negative, we will talk about compactifications and just a little bit about ultrafilters. In fact $\beta\mathbb{N}$ the Stone-Çech compactifcations of the integers (or the set of ultrafilters over $\mathbb{N}$) is a counterexample.

Efimov's Problem asks whether $\beta\mathbb{N}$ is the counterexample to Question 1; that is:

Question 2 (a.k.a. Efimov's Problem): Does every compact space contain either:

• An infinite convergent sequence.
• A homeomorphic copy of $\beta\mathbb{N}$?

In this talk we will see what is going on with the answer to Question 2 that although is not positive, perhaps is not negative either.

November 19, 2018

Sequence Based Clustering Corporate Credit Rating

The K-means algorithm has been applied in many different fields. K-means popularity can be attributed to its simple and intuitive implementation. This simplicity comes at a price, however, as choosing the appropriate number of clusters before hand remains one of the draw backs of the algorithm. In this talk we will review some methods to determine the number of clusters, $K$. We will conclude with how one can apply this algorithm to credit rating data and how effective it might be.

November 12, 2018

Distribution of Values of class numbers over Function Fields

In this talk I will describe a classic problem in number theory: studying class numbers for quadratic extensions of $\mathbb{Q}$ and how to adapt this problem to one that makes sense over function fields.

November 5, 2018

Free probability theory and its combinatorial aspects

In this talk, we will focus probabilistic methods in the non-commutative setting, within the framework of the theory of Operator Algebras. We will discuss a generalization of the notion of a probability space and the kind of independence that is meaningful fo random variables which do not necessarily commute. Finally, we will present the combinatorial characterization of free independence for families of algebras.

September 24, 2018

Title: Welcome Back: Kick-Off Problem Session

To commemorate our first week back, we will have our Second Annual Kick-Off Problem Session. We will have audience members introduce a problem they are interested in studying. Each person will take only 5 minutes to give the problem statement and some of the tools used. This session will act as a preview for what may appear in the upcoming weeks. Afterward we will try to fill up the time slots with volunteer speakers.

October 1, 2018

Physiology Based Lie Detection

Lying, or the act of deceit, is a ubiquitous human talent. Lying typically results in immense social, political, economic, and emotional costs for the deceived, as well as for the deceiver. Due to these costs, there has long been intensive interest in lie detection. In this talk, we will begin with a review of a variety of machine assisted lie detection methodologies which make use of uncontrollable physiological responses to stress induced by lying. We will discuss an exciting new methodology using trans-dermal optical imaging technology to track changes in facial blood flow from video sequences, as well as many of the mathematical challenges associated with detecting deception as it occurs. These challenges reveal deep connections with statistical change point detection methods, time series analysis, and optimal stopping problems in stochastic control, of which a general overview will be provided.

October 15, 2018

Speaker: Massoud Ataei (York U)

An N-th order tensor is a multidimensional array that is defined on a field equipped with the tensor product of N vector spaces. In this talk, I will first define the outer, inner, Hadamard and n-mode products of the tensors and discuss their basic properties. Then, the Canonical Polyadic Decomposition (CPD) will be presented where a tensor is decomposed as a sum of several rank-1 tensors. As an illustration, I will further provide my recent results on applications of the CPD to S&P500 data analysis.

October 22, 2018

Building in properties rather than uncovering them.

In this talk we will focus on a clever topological space constructed by Eric K. van Douwen in 1992. The real numbers is the underlying set of this space. We will discuss the relationship of this example and my current research (interests). No knowledge of topology is assumed. Definitions will be provided.

October 29, 2018

An example of Cluster Analysis in Sports Analytics

In this talk we will focus on the neural network based projective algorithm (PART) and on a possible application to Sports Analytics. The question that motivates the work is: Is it possible fo ra soccer team to buy and sell players under strong financial constraints and keep their competitiveness sufficiently high.

The talk will consist of a basic introduction to clustering, an explanation of the algorithm used (PART), a description of the project proposed and the results obtained.

March 1, 2018

Title: New bounds for $\psi(x;q,a)$

Let $a,q$ be relatively prime integers. Then consider \begin{equation} \psi(x;q,a)=\sum_{\substack{n\le x\\ n\equiv a(\bmod q)}}\Lambda(n). \end{equation} We discuss explicit bounds for $\psi(x;q,a)$, which provide an extension and an improvement over the bounds given by Ramaré and Rumely in 1996. This article introduces three novel pieces to the argument.

February 15, 2018

Title: The Laplace transform of the lognormal distribution

Some integral transforms of the lognormal distribution, such as the Laplace and Fourier transforms, have no known closed form. Several approximations and numerical methods for computing the Laplace transform of the lognormal distribution (LTLD) have been proposed in the literature. The majority of these methods are only valid for complex arguments with nonnegative real part (at best).

In this talk we will explore the analytic continuation of the LTLD to $\mathbb{C}\setminus(-\infty,0]$. Two integral expressions for the analytic continuation will be presented. There is a well known expression for the characteristic function in the literature that is incorrect; we will see why it is incorrect and provide the correct expression. An integral expression of the LTLD will be exploited to obtain asymptotic series approximations. We will discuss the computation of the LTLD by way of numerical integration and series approximations.

If time permits, we will discuss applications like computing the density of a sum of independent lognormals, or computing the density of the Thorin measure of a lognormal distribution.

February 8, 2018

Title: Regular Toirodal Polytopes and Hypertopes

Toroidal polytopes can be built by twisting a portion of a Euclidean tesselation into a torus by identifying translation vectors. Hypertopes are generalizations of polytopes which no longer require vertices, edges, etc... to be ordered. Hypertopes can "tile" Euclidean space, and so we need only find which vectors generate toroidal Hypertopes.

February 1, 2018

Title: Card Shuffling and Hopf Algebras

Riffle shuffling is one of the most popular techniques to randomize a deck of cards. One immediate question follows: how many times do I have to shuffle so that my deck is random? I will talk about this question using the language of Hopf algebras.

January 25, 2018

Title: Honey Bees and Equations of Sociality

It’s no secret that honey bees are dying. The causes of this phenomenon is multi- facted and dependent on complex interactions between environment, pathogens and the structure of of honey bee colonies. Mathematical models help to discern how stresses on a honey bee colony may cause declines in the population, and can offer proposed avenues for conservation and re-population efforts. In this talk, I will painstakingly develop a base model for honey bee colony dynamics and show how such a model can help answer some open questions in biology. Once resolved, we will venture into more current research in the field.

January 18, 2018

Winter Term Kick-off Problem Session

Welcome back to the Winter Term, and the Winter 2018 Series of the Left to the Reader Seminar! To start things off this term, we will have an open problem session again, where audience members will share the problems they are working on, or are interested in working on. Each person will take five minutes to share their current interests to a general mathematical audience, and explain some of the tools most commonly used in their field. Afterward, we will fill up the time slots of the coming term. Hope to see you all there!

November 30, 2017

Title: BlockChain Madness! An introduction to the world of Bitcoin and decentralized networks

Blockchain is a decentralized transaction and data management technology developed first for Bitcoin cryptocurrency in 2008. The technology is based on a distributed public ledger structure, in which the ledger is not owned or controlled by one central authority, transactions are immutable and transparent, and eliminates the so-called double spend problem. For these reasons, blockchain technology is expected to revolutionize industry and commerce and drive economic change on a global scale. It has potential to empower people in developing countries with recognized identity or asset ownership, provide financial independence, and avert financial crises.

Bitcoin, the decentralized peer-to-peer digital currency, is the most popular application of blockchain technology. The digital currency itself is highly controversial but the underlying blockchain technology has worked flawlessly and found wide range of applications in business, politics, health, and society at large. For example, NASDAQ in partnership with Chain are working on shares trading using blockchain; Verisart is using blockchain to verify art prices and encoding copyrights of art work; and ShoCard encodes and stores personal information regarding identity on the blockchain. In this talk, I will explore blockchain technology as a significant source of disruptive innovations and the possible paradigm shift to a democratic scalable digital economy. If time permits, I will provide an overview of the mathematical foundation of blockchain development, specifically, finite fields and elliptic curve cryptography.

November 23, 2017

Title: The limiting distribution of composite likelihood ratio test under non-standard conditions

Full likelihood estimation is a well known and traditional method for parameter estimation. However, correlation and high dimensionality in data often could make the computation of maximum likelihood very intensive and prohibitive. Composite likelihood estimation was introduced as an alternative to the full likelihood that by using sub-densities instead of the joint densities makes the work more feasible. Therefore, I focus on finding the distribution of a composite likelihood version of a hypothesis test which has more complicated form than the full likelihood one.

Full likelihood estimation is a well known and traditional method for parameter estimation. However, correlation and high dimensionality in data often could make the computation of maximum likelihood very intensive and prohibitive. Composite likelihood estimation was introduced as an alternative to the full likelihood that by using sub-densities instead of the joint densities makes the work more feasible. Therefore, I focus on finding the distribution of a composite likelihood version of a hypothesis test which has more complicated form than the full likelihood one.

November 16, 2017

Title: Modelling WNV epidemics in Emilia-Romagna

West Nile Virus (WNV) has been identified for the first time in Italy in 1998, and more continuously since 2008 with a total of 173 neurological human cases between 2008 and 2015 and has become endemic also in Canada - especially in the the Ontario region. The region Emilian Romagna has set up since 2009 a systematic program of mosquito and corvids (known to be among the most competent bird species for WNV) trapping and testing. Data collected through this program has been analysed through a mathematical model in order to understand the main drivers of the observed dynamics. Our results showed that including a seasonal shift in mosquito feeding behaviour (and not keeping it constant) makes model outputs much more consistent with observed data.

In this talk, I would first like to disucss with you about some of the key facts related to West Nile Virues and introduce the notion of compartmental models in disease modelling.. In the second part, I would like to show parts of our results including the ODE model for the mosquito-corvid dynamics and some Markov Chain Monte Carlo (MCMC) methods to estimate the parameters required.

November 9, 2017

Title: A Combinatorial counting construction on $\omega_{1}$

For this talk I will discuss an introductory overview of a recent work with Fulgencio Lopez.

We show that adding at least $\omega_2$ Cohen reals adds a capturing construction scheme. We study the weaker notion of $n$-capturing construction scheme, and show it is consistent to have an $n$-capturing construction scheme, but no $(n+1)$-capturing construction schemes. We also study the relation of n-capturing with the $m$-Knaster hierarchy, and show that $\text{MA}_{\omega_1}(K_m)$ and $n$-capturing are independent when $n$ is at most $m$, and incompatible if $n>m$.

November 2, 2017

Nonlinear statistical filtering for noise removal in radar target tracking

The common usage of the word "filter" refers to a device that removes unwanted components from a mixture, such as a cigarette filter reducing the number of fine particles inhaled with tobacco smoke. Similarly, in signal processing, a filter refers to a process that removes unwanted components from a signal. Often, a signal processing filter removes a range of frequencies from a signal. It "filters out" unwanted frequency components. A statistically defined filter, however, filters out noise from a noisy signal. The first statistically defined filter to be described was the Wiener filter, developed by Norbert Wiener during the 1940's. It paved the way for other statistically defined filters to be introduced, including the Kalman filter, which is the focus of this talk. I will start with a brief description of the Wiener filter, then describe and show an implementation of the (linear) Kalman filter. Next, I will show three nonlinear filters based on the Kalman filter: the extended Kalman filter, the second order nonlinear filter, and the Monte Carlo simulation filter. Finally, I will show results of a simulation study comparing these filters applied to a nonlinear radar target tracking problem.

October 19, 2017

Explicit results on Chebyshev functions: A prime counting adventure

Consider the function $\pi(x)=\{n\le x : n \text{ is prime}\}$. Legendre conjectured that $\pi(x)\sim \frac{x}{\log x}$. Nearly 100 years later, in 1896, Hadamard and de la Vallée Poussin proved an equivalent statement by considering the weighted prime counting function $\psi(x)$. In 1962, Rosser and Schoenfeld gave a method to explicitly estimate the error term in the approximation of $\psi(x)$. This result relies on information concerning the non-trivial zeros of the Riemann zeta function $\zeta(s)$ and subsequent numerical improvements to this information also translated into improved estimates for $\psi(x)$.

In this talk, we present various new explicit methods such as introducing some smooth weights and establishing some zero density estimates for the Riemann zeta function. Additionally, we will discuss results using these new techniques for finding primes in short intervals and the distribution of primes in arithmetic progressions.

October 12, 2017

Problems on D-spaces, Menger spaces and the star versions of the Menger property

A topological space $X$ is a $D$-space if for every neighborhood assignment $\{N(x):x \in X\}$ (that is, $N(x)$ is an open neighborhood of $x$ for each $x \in X$), there is a closed discrete subset $D$ of $X$ such that $\{N(x):x \in D\}$ is a cover of $X$. A topological space $X$ is Menger if for each sequence $\{\mathcal{U}_n:n\in\omega\}$ of open covers of $X$, there is a sequence $\{\mathcal{V}_n:n\in\omega\}$ such that for each $n\in\omega$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and $\{\bigcup\mathcal{V}_n:n\in\omega\}$ is an open cover of $X$. Todd Eisworth states: "There are certainly some mathematical questions that arouse the curiosity of almost anyone who comes in contact with them, questions that tempt with the simplicity of of their formulation, tantalize with promises of an elegant solution if only one can look at the problem in just the right way, and taunt with the number of excellent mathematicians who have examined the question in the past and failed to solve it. The theory of $D$-spaces is replete with such questions."

I would like to discuss with you some problems related with $D$-spaces, Menger spaces, and the star versions of the Menger property.

October 5, 2017

Title: Artificial Intelligence and Analysis of Non-stationary Spatial-temporal Data

In this talk, I will first give a brief introduction to the area of Artificial Intelligence (AI) in which the main differences between Strong AI and Weak AI, and some of the mathematical as well as philosophical theories formulated for describing each of these subareas will be discussed. I will further review the advances in Machine Learning research and current state-of-the-art in Deep Learning, Statistical Learning and Tensor Decomposition-based Learning methods. The rest of the talk will focus on the main challenges and difficulties confronted when analyzing the non-stationary spatial-temporal data using the machine learning approach. As a real-world example of such a complex data, electroencephalogram recordings of patients having major depressive disorder will be demonstrated on which I will share some of my recent research achievements. Our proposed framework utilized for analyzing the mentioned data involves the use of various techniques developed in statistics, operations research, machine learning and big data

September 28, 2017

Change-point detection for noisy non-stationary biological signals

Experimentally and clinically collected time series data are often contaminated with significant confounding noise, creating short, non-stationary time series. This noise, due to natural variability and measurement error, poses a challenge to conventional change point detection methods. We proposed a novel, real-time change point detection method for effectively extracting important time points in non-stationary, noisy time series. We validated our approach with three simulated time series, as well as with a physiological data set of simulated labour experiments in fetal sheep. Our methodology allows for the first time the detection of fetal acidemia from changes in the fetus' heart rate variability, rather than traditional invasive methods. We believe that our method demonstrates a first step towards the development of effective, non-invasive real time monitoring during labour from signals which may be easily collected.

September 21, 2017

Kick-off Problem Session

This week we are having audience members introduce a problem they are interested in studying. Each person will take only 5 minutes to give the problem statement and some of the tools used. This session will act as a preview for what may appear in the upcoming weeks. Afterward we will try to fill up the time slots with volunteer speakers.